A243662 Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.

1, 3, 1, 12, 8, 1, 55, 55, 15, 1, 273, 364, 156, 24, 1, 1428, 2380, 1400, 350, 35, 1, 7752, 15504, 11628, 4080, 680, 48, 1, 43263, 100947, 92169, 41895, 9975, 1197, 63, 1, 246675, 657800, 708400, 396704, 123970, 21560, 1960, 80, 1, 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1

Offset

1,2

Comments

See Novelli-Thibon (2014) for precise definition.

From Tom Copeland, Dec 13 2022: (Start)

The row polynomials are the nonvanishing numerator polynomials generated in the compositional, or Lagrange, inversion in x about the origin of the odd o.g.f. Od1(x,t) = x*(t*(1-x^2)-x^2) / (1-x^2) = t*x - x^3 - x^5 - x^7 - x^9 - ... .

For example, from the Lagrange inversion formula (LIF), the tenth derivative in x of (x/Od1(x,t))^11 / 11! = (1/((t*(1-x^2)-x^2) / (1-x^2)))^11 / 11! at x = 0 is (t^4 + 24*t^3 + 156*t^2 + 364*t + 273) / t^16. These polynomials are also generated by the iterated derivatives ((1/(D Od1(x,t)) D)^n g(x) evaluated at x = 0 where D = d/dx.

An explicit generating function for the polynomials can be obtained by finding the solution of the cubic equation y - t*x - y*x^2 + (1+t)*x^3 = 0 for x in terms of y and t that satisfies y(x=0;t) = 0 = x(y=0;t).

The row polynomials are also the polynomials generated in the compositional inverse of O(x,t) = x / (1+(1+t)x)*(1+x)^2) = x + (-t - 3)*x^2 + (t^2 + 4 t + 6)*x^3 + (-t^3 - 5*t^2 - 10*t - 10)*x^4 + ..., containing the truncated Pascal polynomials of A104712 / A325000.

For example, from the LIF, the third derivative of ((1 + (1+t)*x)*(1+x)^2)^4 / 4! at x = 0 is 55 + 55*t + 15*t^2 + t^3.

A natural refinement of this array was provided in a letter by Isaac Newton in 1676--a set of partition polynomials for generating the o.g.f. of the compositional inverse of the generic odd o.g.f. x + u_1 x^3 + u_2 x^5 + ... in the infinite set of indeterminates u_n. (End)

T(n,k) is the number of noncrossing cacti with n+1 nodes and n+1-k blocks. See A361242. - Andrew Howroyd, Apr 13 2023

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)

Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 10.

Formula

T(n,k) = (binomial(3*n+1,n) * binomial(n,k-1) * binomial(n-1,k-1)) / (binomial(3*n,k-1) * (3*n+1)) = (A001764(n) * A001263(n,k) * k) / binomial(3*n,k-1) for 1 <= k <= n (conjectured). - Werner Schulte, Nov 22 2018

T(n,k) = binomial(3*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n, more generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n and some fixed integer m > 1. - Werner Schulte, Nov 22 2018

G.f.: A(x,y) is the series reversion of x/((1 + x + x*y)*(1 + x)^2). - Andrew Howroyd, Apr 13 2023

Example

Triangle begins:
     1;
     3,    1;
    12,    8,    1;
    55,   55,   15,   1;
   273,  364,  156,  24,  1;
  1428, 2380, 1400, 350, 35, 1;
  ...

Mathematica

T[m_][n_, k_] := Binomial[(m + 1) n + 1 - k, n - k] Binomial[n, k - 1]/n;
Table[T[2][n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

Prog

(PARI)
T(n)=[Vecrev(p) | p<-Vec(serreverse(x/((1+x+x*y)*(1+x)^2) + O(x*x^n)))]
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Apr 13 2023

Crossrefs

Cf. A001764, A001263, A243663 (m=3).

Row sums give A003168.

Row reversed triangle is A102537.

Sequence in context: A133366 A049458 A143492 * A062139 A156366 A144353

Adjacent sequences: A243659 A243660 A243661 * A243663 A243664 A243665

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 13 2014

Extensions

Data and Example (T(2,2) and T(5,3)) corrected and more terms added by Werner Schulte, Nov 22 2018

Status

approved